Open App

JEE Exam > JEE Notes > Mathematics (Maths) for JEE Main & Advanced > Revision Notes: Limits & Derivatives

Limits & Derivatives Class 11 Notes Maths Chapter 12

Download, print and study this document offline

Please wait while the PDF view is loading

 Page 1


 
KEY POINTS
?
x c
lim
?
f(x) = l if and only if
 
? lim ( ) lim ( )
? ?
? ?
? ?
x c x c
f x f x l
?
x c
lim
?
? ? ? ? ? ? ? where ? is a fixed real number. .
 
?
x c
lim
?
x
n
 = c
n
, for all n ? N
 
?
x c
lim
?
f(x) = f(c), where f(x) is a real polynomial in x.
Algebra of limits 
Let f, g be two functions such that
x c
lim
?
f(x) = l and
x c
lim
?
g(x) = m, then
?
x c
lim
?
[ ? f(x)] = ?
x c
lim
?
f(x)
 
= ? l for all ? ? ? R
Page 2


 
KEY POINTS
?
x c
lim
?
f(x) = l if and only if
 
? lim ( ) lim ( )
? ?
? ?
? ?
x c x c
f x f x l
?
x c
lim
?
? ? ? ? ? ? ? where ? is a fixed real number. .
 
?
x c
lim
?
x
n
 = c
n
, for all n ? N
 
?
x c
lim
?
f(x) = f(c), where f(x) is a real polynomial in x.
Algebra of limits 
Let f, g be two functions such that
x c
lim
?
f(x) = l and
x c
lim
?
g(x) = m, then
?
x c
lim
?
[ ? f(x)] = ?
x c
lim
?
f(x)
 
= ? l for all ? ? ? R
 
?
x c
lim
?
[f(x) ± g(x)] = 
x c
lim
?
f(x) ± 
x c
lim
?
g(x) = l ± m
?
x c
lim
?
[f(x).g(x)] = 
x c
lim
?
f(x). 
x c
lim
?
g(x) = lm
 
?
x c
f(x)
lim
g(x)
?
 = 
x c
x c
lim f(x)
lim g(x) m
?
?
?
l
, m ? 0 g(x) ? 0
?
x c
1
lim
f(x)
?
=
x c
1
lim f(x)
?
=
1
l
provided l ? 0 f(x) ? 0
?
x c
lim
?
[(f(x)]
n
=
n
x c
lim f(x)
?
? ? ? ?
? ? ? ?
?
?
?
? ? ? ?
? ?
? ?
= l
n
, for all n ? N
Some important theorems on limits 
?
x 0
lim
?
?
f(x) =
x 0
lim
?
?
f(–x)
 
?
n n
x a
x a
lim
x a
?
?
?
= na
n – 1
?
x 0
sin x
lim
x
?
=1 where x is measured in radians.
?
x 0 x 0
tan x cos x
lim 1 Note that lim 1
x x
? ?
? ?
? ?
? ?
? ?
 
?
x 0
1 cos x
lim 0
x
?
?
?
?
x
x 0
e 1
lim 1
x
?
?
?
Page 3


 
KEY POINTS
?
x c
lim
?
f(x) = l if and only if
 
? lim ( ) lim ( )
? ?
? ?
? ?
x c x c
f x f x l
?
x c
lim
?
? ? ? ? ? ? ? where ? is a fixed real number. .
 
?
x c
lim
?
x
n
 = c
n
, for all n ? N
 
?
x c
lim
?
f(x) = f(c), where f(x) is a real polynomial in x.
Algebra of limits 
Let f, g be two functions such that
x c
lim
?
f(x) = l and
x c
lim
?
g(x) = m, then
?
x c
lim
?
[ ? f(x)] = ?
x c
lim
?
f(x)
 
= ? l for all ? ? ? R
 
?
x c
lim
?
[f(x) ± g(x)] = 
x c
lim
?
f(x) ± 
x c
lim
?
g(x) = l ± m
?
x c
lim
?
[f(x).g(x)] = 
x c
lim
?
f(x). 
x c
lim
?
g(x) = lm
 
?
x c
f(x)
lim
g(x)
?
 = 
x c
x c
lim f(x)
lim g(x) m
?
?
?
l
, m ? 0 g(x) ? 0
?
x c
1
lim
f(x)
?
=
x c
1
lim f(x)
?
=
1
l
provided l ? 0 f(x) ? 0
?
x c
lim
?
[(f(x)]
n
=
n
x c
lim f(x)
?
? ? ? ?
? ? ? ?
?
?
?
? ? ? ?
? ?
? ?
= l
n
, for all n ? N
Some important theorems on limits 
?
x 0
lim
?
?
f(x) =
x 0
lim
?
?
f(–x)
 
?
n n
x a
x a
lim
x a
?
?
?
= na
n – 1
?
x 0
sin x
lim
x
?
=1 where x is measured in radians.
?
x 0 x 0
tan x cos x
lim 1 Note that lim 1
x x
? ?
? ?
? ?
? ?
? ?
 
?
x 0
1 cos x
lim 0
x
?
?
?
?
x
x 0
e 1
lim 1
x
?
?
?
 
?
x
e
x 0
a 1
lim log a
x
?
?
?
?
x 0
log(1 x)
lim 1
x
?
?
?
?
? ?
1 x
x 0
lim 1 x e
?
? ?
? ? s in
cos
d x
x
dx
?
 
? ? cos
s in
d x
x
dx
? ?
 
? ?
2
ta n
se c
d x
x
dx
?
 
? ?
2
cot
cos
d x
ec x
dx
? ?
 
? ? se c
se c ta n
d x
x x
dx
?
? ? cos
cos .cot
d ec x
ec x x
dx
? ?
? ?
1
.
n
n
d x
n x
dx
?
? 
Page 4


 
KEY POINTS
?
x c
lim
?
f(x) = l if and only if
 
? lim ( ) lim ( )
? ?
? ?
? ?
x c x c
f x f x l
?
x c
lim
?
? ? ? ? ? ? ? where ? is a fixed real number. .
 
?
x c
lim
?
x
n
 = c
n
, for all n ? N
 
?
x c
lim
?
f(x) = f(c), where f(x) is a real polynomial in x.
Algebra of limits 
Let f, g be two functions such that
x c
lim
?
f(x) = l and
x c
lim
?
g(x) = m, then
?
x c
lim
?
[ ? f(x)] = ?
x c
lim
?
f(x)
 
= ? l for all ? ? ? R
 
?
x c
lim
?
[f(x) ± g(x)] = 
x c
lim
?
f(x) ± 
x c
lim
?
g(x) = l ± m
?
x c
lim
?
[f(x).g(x)] = 
x c
lim
?
f(x). 
x c
lim
?
g(x) = lm
 
?
x c
f(x)
lim
g(x)
?
 = 
x c
x c
lim f(x)
lim g(x) m
?
?
?
l
, m ? 0 g(x) ? 0
?
x c
1
lim
f(x)
?
=
x c
1
lim f(x)
?
=
1
l
provided l ? 0 f(x) ? 0
?
x c
lim
?
[(f(x)]
n
=
n
x c
lim f(x)
?
? ? ? ?
? ? ? ?
?
?
?
? ? ? ?
? ?
? ?
= l
n
, for all n ? N
Some important theorems on limits 
?
x 0
lim
?
?
f(x) =
x 0
lim
?
?
f(–x)
 
?
n n
x a
x a
lim
x a
?
?
?
= na
n – 1
?
x 0
sin x
lim
x
?
=1 where x is measured in radians.
?
x 0 x 0
tan x cos x
lim 1 Note that lim 1
x x
? ?
? ?
? ?
? ?
? ?
 
?
x 0
1 cos x
lim 0
x
?
?
?
?
x
x 0
e 1
lim 1
x
?
?
?
 
?
x
e
x 0
a 1
lim log a
x
?
?
?
?
x 0
log(1 x)
lim 1
x
?
?
?
?
? ?
1 x
x 0
lim 1 x e
?
? ?
? ? s in
cos
d x
x
dx
?
 
? ? cos
s in
d x
x
dx
? ?
 
? ?
2
ta n
se c
d x
x
dx
?
 
? ?
2
cot
cos
d x
ec x
dx
? ?
 
? ? se c
se c ta n
d x
x x
dx
?
? ? cos
cos .cot
d ec x
ec x x
dx
? ?
? ?
1
.
n
n
d x
n x
dx
?
? 
 
? ?
x
x
d e
e
dx
?
? ?
.log
x
x
d a
a a
dx
?
? ? log 1
e
d x
dx x
?
? ?
0
d constant
dx
?
 
Laws of Logarithm 
? ? ? log log log
e e e
A B AB ? ?
?
log log log
e e e
A
A B
B
? ?
? ?
? ?
? ?
? log log
m
e e
A m A ? 
? log 1 0
a
?
? If log then ? ?
x
B
A x B A
Let y = f(x) be a function defined in some neighbourhood of the 
point ‘a’. Let P(a, f(a)) and Q(a + h, f (a + h)) are two points on 
the graph of f(x) where h is very small and 0 h ? . 
Page 5


 
KEY POINTS
?
x c
lim
?
f(x) = l if and only if
 
? lim ( ) lim ( )
? ?
? ?
? ?
x c x c
f x f x l
?
x c
lim
?
? ? ? ? ? ? ? where ? is a fixed real number. .
 
?
x c
lim
?
x
n
 = c
n
, for all n ? N
 
?
x c
lim
?
f(x) = f(c), where f(x) is a real polynomial in x.
Algebra of limits 
Let f, g be two functions such that
x c
lim
?
f(x) = l and
x c
lim
?
g(x) = m, then
?
x c
lim
?
[ ? f(x)] = ?
x c
lim
?
f(x)
 
= ? l for all ? ? ? R
 
?
x c
lim
?
[f(x) ± g(x)] = 
x c
lim
?
f(x) ± 
x c
lim
?
g(x) = l ± m
?
x c
lim
?
[f(x).g(x)] = 
x c
lim
?
f(x). 
x c
lim
?
g(x) = lm
 
?
x c
f(x)
lim
g(x)
?
 = 
x c
x c
lim f(x)
lim g(x) m
?
?
?
l
, m ? 0 g(x) ? 0
?
x c
1
lim
f(x)
?
=
x c
1
lim f(x)
?
=
1
l
provided l ? 0 f(x) ? 0
?
x c
lim
?
[(f(x)]
n
=
n
x c
lim f(x)
?
? ? ? ?
? ? ? ?
?
?
?
? ? ? ?
? ?
? ?
= l
n
, for all n ? N
Some important theorems on limits 
?
x 0
lim
?
?
f(x) =
x 0
lim
?
?
f(–x)
 
?
n n
x a
x a
lim
x a
?
?
?
= na
n – 1
?
x 0
sin x
lim
x
?
=1 where x is measured in radians.
?
x 0 x 0
tan x cos x
lim 1 Note that lim 1
x x
? ?
? ?
? ?
? ?
? ?
 
?
x 0
1 cos x
lim 0
x
?
?
?
?
x
x 0
e 1
lim 1
x
?
?
?
 
?
x
e
x 0
a 1
lim log a
x
?
?
?
?
x 0
log(1 x)
lim 1
x
?
?
?
?
? ?
1 x
x 0
lim 1 x e
?
? ?
? ? s in
cos
d x
x
dx
?
 
? ? cos
s in
d x
x
dx
? ?
 
? ?
2
ta n
se c
d x
x
dx
?
 
? ?
2
cot
cos
d x
ec x
dx
? ?
 
? ? se c
se c ta n
d x
x x
dx
?
? ? cos
cos .cot
d ec x
ec x x
dx
? ?
? ?
1
.
n
n
d x
n x
dx
?
? 
 
? ?
x
x
d e
e
dx
?
? ?
.log
x
x
d a
a a
dx
?
? ? log 1
e
d x
dx x
?
? ?
0
d constant
dx
?
 
Laws of Logarithm 
? ? ? log log log
e e e
A B AB ? ?
?
log log log
e e e
A
A B
B
? ?
? ?
? ?
? ?
? log log
m
e e
A m A ? 
? log 1 0
a
?
? If log then ? ?
x
B
A x B A
Let y = f(x) be a function defined in some neighbourhood of the 
point ‘a’. Let P(a, f(a)) and Q(a + h, f (a + h)) are two points on 
the graph of f(x) where h is very small and 0 h ? . 
 
Slope of PQ = 
? ? ? ? f a h f a
h
? ?
If 0 h ? , point Q approaches to P and the line PQ becomes a 
tangent to the curve at point P. 
? ? ? ?
0
lim
h
f a h f a
h
?
? ?
 (if exists) is called derivative of f(x) at the 
point ‘a’. It is denoted by f’(a). 
Algebra of derivatives 
?
? ? ? ? ? ? ? ?
.
d d
cf x c f x
dx dx
?
where c is a constant 
?
? ? ? ? ? ? ? ? ? ? ? ? ? ?
d d d
f x g x f x g x
dx dx dx
? ? ?
?
? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?
. .
d d d
f x g x f x g x g x f x
dx dx dx
? ?

	Mathematics (Maths) for JEE Main & Advanced 209 videos\|443 docs\|143 tests

Mathematics (Maths) for JEE Main & Advanced

209 videos|443 docs|143 tests

Join Course for Free

Top Courses for JEE

View all

FAQs on Limits & Derivatives Class 11 Notes Maths Chapter 12

1. What is the concept of limits in calculus?

Ans. In calculus, limits are used to determine the behavior of a function as it approaches a certain value or as the input approaches infinity or negative infinity. It helps us understand how a function behaves near a particular point.

2. How do you find the limit of a function algebraically?

Ans. To find the limit of a function algebraically, you can try to simplify the expression or use algebraic manipulation techniques such as factoring or rationalizing the numerator or denominator. Sometimes, you may need to apply limit laws or use special limit theorems to evaluate the limit.

3. What is the significance of derivatives in calculus?

Ans. Derivatives in calculus represent the rate at which a function is changing at any given point. They provide valuable information about the slope of a function, the direction of its graph, and can be used to find maximum and minimum points. Derivatives are essential in many areas of science, engineering, and economics.

4. How do you find the derivative of a function?

Ans. To find the derivative of a function, you can use differentiation techniques such as the power rule, product rule, quotient rule, or chain rule. These rules allow you to find the rate of change of a function with respect to its independent variable.

5. What is the relationship between limits and derivatives?

Ans. The concept of a derivative is closely related to the concept of a limit. In fact, the derivative of a function at a point is defined as the limit of the average rate of change of the function as the interval approaches zero. Limits are used in calculus to study the behavior of functions, while derivatives provide information about the rate of change of a function at a specific point.

Related Exams

JEE Grade 9 Grade 10 Grade 11 Grade 12

About this Document

	4.70/5 Rating
	Nov 21, 2024 Last updated

Document Description: Revision Notes: Limits & Derivatives for JEE 2024 is part of Mathematics (Maths) for JEE Main & Advanced preparation. The notes and questions for Revision Notes: Limits & Derivatives have been prepared according to the JEE exam syllabus. Information about Revision Notes: Limits & Derivatives covers topics like and Revision Notes: Limits & Derivatives Example, for JEE 2024 Exam. Find important definitions, questions, notes, meanings, examples, exercises and tests below for Revision Notes: Limits & Derivatives.

Introduction of Revision Notes: Limits & Derivatives in English is available as part of our Mathematics (Maths) for JEE Main & Advanced for JEE & Revision Notes: Limits & Derivatives in Hindi for Mathematics (Maths) for JEE Main & Advanced course. Download more important topics related with notes, lectures and mock test series for JEE Exam by signing up for free. JEE: Limits & Derivatives Class 11 Notes Maths Chapter 12

Description

Full syllabus notes, lecture & questions for Limits & Derivatives Class 11 Notes Maths Chapter 12 - JEE | Plus excerises question with solution to help you revise complete syllabus for Mathematics (Maths) for JEE Main & Advanced | Best notes, free PDF download

Information about Revision Notes: Limits & Derivatives

In this doc you can find the meaning of Revision Notes: Limits & Derivatives defined & explained in the simplest way possible. Besides explaining types of Revision Notes: Limits & Derivatives theory, EduRev gives you an ample number of questions to practice Revision Notes: Limits & Derivatives tests, examples and also practice JEE tests

	Mathematics (Maths) for JEE Main & Advanced 209 videos\|443 docs\|143 tests

Mathematics (Maths) for JEE Main & Advanced

209 videos|443 docs|143 tests

Join Course for Free

Download as PDF

Explore Courses for JEE exam

Top Courses for JEE

Explore Courses

Signup for Free!

Signup to see your scores go up within 7 days! Learn & Practice with 1000+ FREE Notes, Videos & Tests.

Start learning for Free

10M+ students study on EduRev

MCQs

Exam

ppt

Limits & Derivatives Class 11 Notes Maths Chapter 12

Semester Notes

Extra Questions

study material

past year papers

Important questions

Viva Questions

pdf

Sample Paper

video lectures

mock tests for examination

practice quizzes

Summary

Objective type Questions

shortcuts and tricks

Limits & Derivatives Class 11 Notes Maths Chapter 12

Free

Previous Year Questions with Solutions

;

Additional Information about Revision Notes: Limits & Derivatives for JEE Preparation

Revision Notes: Limits & Derivatives Free PDF Download

The Revision Notes: Limits & Derivatives is an invaluable resource that delves deep into the core of the JEE exam. These study notes are curated by experts and cover all the essential topics and concepts, making your preparation more efficient and effective. With the help of these notes, you can grasp complex subjects quickly, revise important points easily, and reinforce your understanding of key concepts. The study notes are presented in a concise and easy-to-understand manner, allowing you to optimize your learning process. Whether you're looking for best-recommended books, sample papers, study material, or toppers' notes, this PDF has got you covered. Download the Revision Notes: Limits & Derivatives now and kickstart your journey towards success in the JEE exam.

Importance of Revision Notes: Limits & Derivatives

The importance of Revision Notes: Limits & Derivatives cannot be overstated, especially for JEE aspirants. This document holds the key to success in the JEE exam. It offers a detailed understanding of the concept, providing invaluable insights into the topic. By knowing the concepts well in advance, students can plan their preparation effectively. Utilize this indispensable guide for a well-rounded preparation and achieve your desired results.

Revision Notes: Limits & Derivatives

Revision Notes: Limits & Derivatives Notes offer in-depth insights into the specific topic to help you master it with ease. This comprehensive document covers all aspects related to Revision Notes: Limits & Derivatives. It includes detailed information about the exam syllabus, recommended books, and study materials for a well-rounded preparation. Practice papers and question papers enable you to assess your progress effectively. Additionally, the paper analysis provides valuable tips for tackling the exam strategically. Access to Toppers' notes gives you an edge in understanding complex concepts. Whether you're a beginner or aiming for advanced proficiency, Revision Notes: Limits & Derivatives Notes on EduRev are your ultimate resource for success.

Revision Notes: Limits & Derivatives JEE Questions

The "Revision Notes: Limits & Derivatives JEE Questions" guide is a valuable resource for all aspiring students preparing for the JEE exam. It focuses on providing a wide range of practice questions to help students gauge their understanding of the exam topics. These questions cover the entire syllabus, ensuring comprehensive preparation. The guide includes previous years' question papers for students to familiarize themselves with the exam's format and difficulty level. Additionally, it offers subject-specific question banks, allowing students to focus on weak areas and improve their performance.

Study Revision Notes: Limits & Derivatives on the App

Students of JEE can study Revision Notes: Limits & Derivatives alongwith tests & analysis from the EduRev app, which will help them while preparing for their exam. Apart from the Revision Notes: Limits & Derivatives, students can also utilize the EduRev App for other study materials such as previous year question papers, syllabus, important questions, etc. The EduRev App will make your learning easier as you can access it from anywhere you want. The content of Revision Notes: Limits & Derivatives is prepared as per the latest JEE syllabus.

Education Revolution